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Question 1180101: Maximize and Minimize P=5x+15y<=60
Subject to: x+3y<=60
x+y>=10
x-y<=0
x>=0
x>=0
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
The process for solving a problem like this is well defined. What is it about the process that you need help with?
The instructions for posting your question say you need to show the work you have tried to do on the problem and tell us where you need help. If you just post the problem with no work, it looks as if you just want us to do your work for you.
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I will assume the last constraint is supposed to be y>=0 (although it ends up being irrelevant)....
The way you state the problem, there is in fact no solution. You incorrectly say we are to maximize and minimize "P=5x+15y<=60" subject to the given constraints. However, with the given constraints, P is NEVER less than or equal to 60.
Your statement of the problem should say to maximize and minimize "P=5x+15y" subject to the given constraints.
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Step 1: Graph the constraint boundary lines and determine on which side of each line the feasibility region lies.
(1) x+3y<=60 --> y<=(-1/3)x+20
(2) x+y>=10 --> y>=-x+10
(3) x-y<=0 --> y>=x
A graph of the boundary lines: (1) red, (2) green, and (3) blue
The directions of the inequalities for the three constraints tell us that the feasibility region is below the red line and above the green and blue lines, and of course to the right of the y-axis.
Step 2: Determine the coordinates of the corners of the feasibility region.
This involves some basic algebra, solving pairs of linear equations. I will assume that, if you are working on a problem like this, you know how to do that.
Step 3: Evaluate the objective function at each corner of the feasibility region to find its maximum and minimum values.
That also should be an easy task.
I leave the actual work to you. If you need help finishing the problem, re-post the problem showing what work you have done and telling what you still need help with.
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One comment about the solution....
The slope of the objective function is the same as the slope of the constraint (1) boundary line. That means the maximum value of the objective function will be obtained anywhere along the constraint (1) boundary line that is part of the feasibility region; you will not get a maximum value at a single corner of the feasibility region.
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